# mg.metric geometry – Can each third-dimensional convex corpse breathe trapped by a tetrahedral cage? Answer

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mg.metric geometry – Can each third-dimensional convex corpse breathe trapped by a tetrahedral cage?

Although the title query is pretty unambiguous, I give all pertinent definitions:

$$bullet$$ A subset $$C$$ of $$mathbb{R}^n$$ is an $$n$$-dimensional convex corpse if $$C$$ is convex, compact, and has non-empty inside.

$$bullet$$ A polyhedral cage $$P^{(1)}$$ in $$mathbb{R}^n$$ is the union of all edges (i.e., the 1-skeleton) of a convex $$n$$-dimensional polyhedron $$P$$. In specific, a tetrahedral cage is the union of the six edges of some tetrahedron in $$mathbb{R}^3$$.

$$bullet$$ A convex $$3$$-dimensional corpse $$C$$ is trapped by the tetrahedral cage $$T^{(1)}$$, that’s, by the $$1$$-skeleton of the tetrahedron $$T$$, if the cage is mounted (immobile), and if for each steady inflexible movement (rotations allowed) $$f_t(C); 0le tle 1$$, both $$f_t(C)$$ intersects $$T$$ for each $$tin [0,1]$$ or $$T^{(1)}$$ accommodates an inside level of $$f_t(C)$$ for some $$t_0in [0,1]$$.
In different phrases, $$C$$ can not breathe moved arbitrarily removed from $$T$$ whereas, throughout the gross movement, avoiding the cage’s bars penetrating $$C$$‘s inside.

$$bullet$$ Remark. In dimension $$n>3$$, one can deem analogous questions, with quite a lot of sorts of a simplicial cage, by taking the $$i$$-skeleton of a convex $$n$$-dimensional simplex, with $$1le ile n-2$$.

A trifling instance: A ball is trapped by the $$1$$-skeleton of the common tetrahedron edge-tangent to the ball. And, clearly, each convex corpse can breathe trapped by some polyhedral cage.

For an reader, I imply a number of much less trifling workouts: every of the next convex our bodies can breathe trapped by a tetrahedral cage: the dice; the round cylinder of any (finite) peak; the round cone; the common tetrahedron.

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